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Theorem r0cld 23747
Description: The analogue of the T1 axiom (singletons are closed) for an R0 space. In an R0 space the set of all points topologically indistinguishable from 𝐴 is closed. (Contributed by Mario Carneiro, 25-Aug-2015.)
Hypothesis
Ref Expression
kqval.2 𝐹 = (𝑥𝑋 ↦ {𝑦𝐽𝑥𝑦})
Assertion
Ref Expression
r0cld ((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Fre ∧ 𝐴𝑋) → {𝑧𝑋 ∣ ∀𝑜𝐽 (𝑧𝑜𝐴𝑜)} ∈ (Clsd‘𝐽))
Distinct variable groups:   𝑥,𝑜,𝑦,𝑧,𝐴   𝑜,𝐽,𝑥,𝑦,𝑧   𝑜,𝐹,𝑧   𝑜,𝑋,𝑥,𝑦,𝑧
Allowed substitution hints:   𝐹(𝑥,𝑦)

Proof of Theorem r0cld
StepHypRef Expression
1 kqval.2 . . . . . 6 𝐹 = (𝑥𝑋 ↦ {𝑦𝐽𝑥𝑦})
21kqffn 23734 . . . . 5 (𝐽 ∈ (TopOn‘𝑋) → 𝐹 Fn 𝑋)
323ad2ant1 1133 . . . 4 ((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Fre ∧ 𝐴𝑋) → 𝐹 Fn 𝑋)
4 fncnvima2 7080 . . . 4 (𝐹 Fn 𝑋 → (𝐹 “ {(𝐹𝐴)}) = {𝑧𝑋 ∣ (𝐹𝑧) ∈ {(𝐹𝐴)}})
53, 4syl 17 . . 3 ((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Fre ∧ 𝐴𝑋) → (𝐹 “ {(𝐹𝐴)}) = {𝑧𝑋 ∣ (𝐹𝑧) ∈ {(𝐹𝐴)}})
6 fvex 6918 . . . . . 6 (𝐹𝑧) ∈ V
76elsn 4640 . . . . 5 ((𝐹𝑧) ∈ {(𝐹𝐴)} ↔ (𝐹𝑧) = (𝐹𝐴))
8 simpl1 1191 . . . . . 6 (((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Fre ∧ 𝐴𝑋) ∧ 𝑧𝑋) → 𝐽 ∈ (TopOn‘𝑋))
9 simpr 484 . . . . . 6 (((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Fre ∧ 𝐴𝑋) ∧ 𝑧𝑋) → 𝑧𝑋)
10 simpl3 1193 . . . . . 6 (((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Fre ∧ 𝐴𝑋) ∧ 𝑧𝑋) → 𝐴𝑋)
111kqfeq 23733 . . . . . . 7 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑧𝑋𝐴𝑋) → ((𝐹𝑧) = (𝐹𝐴) ↔ ∀𝑦𝐽 (𝑧𝑦𝐴𝑦)))
12 eleq2w 2824 . . . . . . . . 9 (𝑦 = 𝑜 → (𝑧𝑦𝑧𝑜))
13 eleq2w 2824 . . . . . . . . 9 (𝑦 = 𝑜 → (𝐴𝑦𝐴𝑜))
1412, 13bibi12d 345 . . . . . . . 8 (𝑦 = 𝑜 → ((𝑧𝑦𝐴𝑦) ↔ (𝑧𝑜𝐴𝑜)))
1514cbvralvw 3236 . . . . . . 7 (∀𝑦𝐽 (𝑧𝑦𝐴𝑦) ↔ ∀𝑜𝐽 (𝑧𝑜𝐴𝑜))
1611, 15bitrdi 287 . . . . . 6 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑧𝑋𝐴𝑋) → ((𝐹𝑧) = (𝐹𝐴) ↔ ∀𝑜𝐽 (𝑧𝑜𝐴𝑜)))
178, 9, 10, 16syl3anc 1372 . . . . 5 (((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Fre ∧ 𝐴𝑋) ∧ 𝑧𝑋) → ((𝐹𝑧) = (𝐹𝐴) ↔ ∀𝑜𝐽 (𝑧𝑜𝐴𝑜)))
187, 17bitrid 283 . . . 4 (((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Fre ∧ 𝐴𝑋) ∧ 𝑧𝑋) → ((𝐹𝑧) ∈ {(𝐹𝐴)} ↔ ∀𝑜𝐽 (𝑧𝑜𝐴𝑜)))
1918rabbidva 3442 . . 3 ((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Fre ∧ 𝐴𝑋) → {𝑧𝑋 ∣ (𝐹𝑧) ∈ {(𝐹𝐴)}} = {𝑧𝑋 ∣ ∀𝑜𝐽 (𝑧𝑜𝐴𝑜)})
205, 19eqtrd 2776 . 2 ((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Fre ∧ 𝐴𝑋) → (𝐹 “ {(𝐹𝐴)}) = {𝑧𝑋 ∣ ∀𝑜𝐽 (𝑧𝑜𝐴𝑜)})
211kqid 23737 . . . 4 (𝐽 ∈ (TopOn‘𝑋) → 𝐹 ∈ (𝐽 Cn (KQ‘𝐽)))
22213ad2ant1 1133 . . 3 ((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Fre ∧ 𝐴𝑋) → 𝐹 ∈ (𝐽 Cn (KQ‘𝐽)))
23 simp2 1137 . . . 4 ((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Fre ∧ 𝐴𝑋) → (KQ‘𝐽) ∈ Fre)
24 simp3 1138 . . . . . 6 ((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Fre ∧ 𝐴𝑋) → 𝐴𝑋)
25 fnfvelrn 7099 . . . . . 6 ((𝐹 Fn 𝑋𝐴𝑋) → (𝐹𝐴) ∈ ran 𝐹)
263, 24, 25syl2anc 584 . . . . 5 ((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Fre ∧ 𝐴𝑋) → (𝐹𝐴) ∈ ran 𝐹)
271kqtopon 23736 . . . . . . 7 (𝐽 ∈ (TopOn‘𝑋) → (KQ‘𝐽) ∈ (TopOn‘ran 𝐹))
28273ad2ant1 1133 . . . . . 6 ((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Fre ∧ 𝐴𝑋) → (KQ‘𝐽) ∈ (TopOn‘ran 𝐹))
29 toponuni 22921 . . . . . 6 ((KQ‘𝐽) ∈ (TopOn‘ran 𝐹) → ran 𝐹 = (KQ‘𝐽))
3028, 29syl 17 . . . . 5 ((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Fre ∧ 𝐴𝑋) → ran 𝐹 = (KQ‘𝐽))
3126, 30eleqtrd 2842 . . . 4 ((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Fre ∧ 𝐴𝑋) → (𝐹𝐴) ∈ (KQ‘𝐽))
32 eqid 2736 . . . . 5 (KQ‘𝐽) = (KQ‘𝐽)
3332t1sncld 23335 . . . 4 (((KQ‘𝐽) ∈ Fre ∧ (𝐹𝐴) ∈ (KQ‘𝐽)) → {(𝐹𝐴)} ∈ (Clsd‘(KQ‘𝐽)))
3423, 31, 33syl2anc 584 . . 3 ((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Fre ∧ 𝐴𝑋) → {(𝐹𝐴)} ∈ (Clsd‘(KQ‘𝐽)))
35 cnclima 23277 . . 3 ((𝐹 ∈ (𝐽 Cn (KQ‘𝐽)) ∧ {(𝐹𝐴)} ∈ (Clsd‘(KQ‘𝐽))) → (𝐹 “ {(𝐹𝐴)}) ∈ (Clsd‘𝐽))
3622, 34, 35syl2anc 584 . 2 ((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Fre ∧ 𝐴𝑋) → (𝐹 “ {(𝐹𝐴)}) ∈ (Clsd‘𝐽))
3720, 36eqeltrrd 2841 1 ((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Fre ∧ 𝐴𝑋) → {𝑧𝑋 ∣ ∀𝑜𝐽 (𝑧𝑜𝐴𝑜)} ∈ (Clsd‘𝐽))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395  w3a 1086   = wceq 1539  wcel 2107  wral 3060  {crab 3435  {csn 4625   cuni 4906  cmpt 5224  ccnv 5683  ran crn 5685  cima 5687   Fn wfn 6555  cfv 6560  (class class class)co 7432  TopOnctopon 22917  Clsdccld 23025   Cn ccn 23233  Frect1 23316  KQckq 23702
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707  ax-rep 5278  ax-sep 5295  ax-nul 5305  ax-pow 5364  ax-pr 5431  ax-un 7756
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ne 2940  df-ral 3061  df-rex 3070  df-reu 3380  df-rab 3436  df-v 3481  df-sbc 3788  df-csb 3899  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-nul 4333  df-if 4525  df-pw 4601  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4907  df-iun 4992  df-br 5143  df-opab 5205  df-mpt 5225  df-id 5577  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-rn 5695  df-res 5696  df-ima 5697  df-iota 6513  df-fun 6562  df-fn 6563  df-f 6564  df-f1 6565  df-fo 6566  df-f1o 6567  df-fv 6568  df-ov 7435  df-oprab 7436  df-mpo 7437  df-map 8869  df-qtop 17553  df-top 22901  df-topon 22918  df-cld 23028  df-cn 23236  df-t1 23323  df-kq 23703
This theorem is referenced by:  nrmr0reg  23758
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